Three prime numbers p, q and r, each less than 20, are such that p - q = q - r . How many distinct possible values can we get for (p + q + r) ?

The question states that p, q, and r are prime numbers less than 20 and follow the equation p minus q equals q minus r. This equation can be rewritten as p plus r equals 2q. This means that q is the exact middle value (arithmetic mean) of p and r, implying that p, q, and r form an arithmetic progression.

First, let us list all prime numbers less than 20: 2, 3, 5, 7, 11, 13, 17, 19.

Now we find all sets of three primes from this list where the common difference is the same:

1. With a common difference of 2: (7, 5, 3). Here, p plus q plus r is 15.
2. With a common difference of 4: (11, 7, 3). Here, p plus q plus r is 21.
3. With a common difference of 6: (17, 11, 5). Here, p plus q plus r is 33.
4. With a common difference of 6: (19, 13, 7). Here, p plus q plus r is 39.

Note that any set involving the number 2 would require the other numbers to be non-primes to maintain the spacing, and other combinations like (17, 13, 9) fail because 9 is not prime.

The four distinct values for the sum (p plus q plus r) are 15, 21, 33, and 39. Since there are exactly 4 distinct possible values, the correct option is A.